The disclosed embodiments relate to the field of collision prediction and avoidance of airborne and spaceborne vehicles. More particularly, the embodiments relate to flight path trajectory conflict prediction and maneuvering avoidance methods for airplanes and spacecraft using parallelepipeds in Mahalanobis space.
The following nomenclature is used herein:
axis12=unit vector from r1 to r2
axis12r=axis12 rotated
axis23=unit vector from r2 to r3
axis23r=axis23 rotated
C3=3×3 positional covariance matrix
dx=off-axis x position
dy=off-axis y position
dz=endpoint adjustment
ECI=Earth centered inertial frame
erf=error function
m=counter upper limit
Mf=final Mahalanobis distance
Mi=initial Mahalanobis distance
n=combined covariance ellipsoid scale factor
OBJ=cross-sectional radius
P=probability
P_S1=Chan's analytical probability approximation
r=radius of torus' cross-sectional (or relative distance vector)
r1=first relative distance point
r2=second relative distance point
r3=third relative distance point
R=radius of torus (or Patera's distance to combined object center)
TCA=time of closest approach
ti=start time
tf=end time
V=swept out volume of collision tube
VNC=Velocity-Normal-Co-Normal frame
X=object's earth-centered x position
y=object's earth-centered y position
z=object's earth-centered z position
α=coefficient defining probability density
φ=object-centric angle in Mahalanobis space
σ=standard deviation
θ=object-centric angle
The assumptions involved in linear collision probability formulation are generally well defined in the prior art. Object collision probability analysis (a.k.a., COLA) is typically conducted with the objects modeled as spheres, thus eliminating the need for attitude information. Their relative motion is considered linear for the encounter by assuming the effect of relative acceleration is dwarfed by that of the velocity. The positional errors are assumed to be zero-mean, Gaussian, uncorrelated, and constant for the encounter. The relative velocity at the point of closest approach is deemed sufficiently large to ensure a brief encounter time and static covariance. The cumulative collision probability P is found by integrating the three-dimensional, Gaussian, relative position density over the volume V (collision tube) that is swept out by the combined hardbody of the two space objects over a specified time interval (ti, tf)
                    P        =                                            1                                                                    8                    ·                                          π                      3                                                                      ·                                  σ                  x                                ·                                  σ                  y                                ·                                  σ                  z                                                      ·                                          ∫                                  ∫                  ∫                                                            V                ⁡                                  (                                                            t                      i                                        ,                                          t                      f                                                        )                                                              ⁢                      exp            ⁡                          [                                                                    -                                          x                      2                                                                            2                    ·                                          (                                              σ                        x                                            )                                                                      +                                                      -                                          y                      2                                                                            2                    ·                                          (                                              σ                        y                                            )                                                                      +                                                      -                                          z                      2                                                                            2                    ·                                          (                                              σ                        z                                            )                                                                                  ]                                ⁢                      ⅆ            x                    ⁢                      ⅆ            y                    ⁢                                          ⁢                      ⅆ            z                                              (        1        )            The probability density in the bracketed section is conveniently represented in the diagonal frame of the position-error covariance matrix. The definition of the integration volume V(ti, tf) is the most complicated aspect of evaluating Equation 1. Coupled with object sizes, the encounter region determines the limits of integration. The encounter region is defined when one object is within a standard deviation (σ) combined covariance ellipsoid shell scaled by a factor of n. This user-defined, three-dimensional, n-σ shell is centered on the primary object; n is typically in the range of 3 to 8 to accommodate conjunction possibilities ranging from 97.070911% to 99.999999%.
Because the covariances are expected to be uncorrelated, they are simply summed to form one, large, combined, covariance ellipsoid 10 that is centered at the primary object. The secondary object 12 passes quickly through this ellipsoid 10 creating a tube-shaped path that is commonly called a collision tube 14. A conjunction occurs if the secondary sphere touches the primary sphere, i.e., when the distance between the two projected object centers is less than the sum of their radii. The radius of this collision tube 14 accommodates all possibilities of the secondary touching the primary by combining the radii of both objects. A plane perpendicular to the relative velocity vector 16 is formed and the combined object and covariance ellipsoid are projected onto this encounter plane 18 as shown in FIG. 1.
As previously stated, the encounter region is defined by an n-σ shell determined by the user to sufficiently account for conjunction possibilities. For short-term encounters, the tube 14 is assumed straight and rapidly traversed, allowing a decoupling of the dimension associated with the tube path (relative velocity). The tube becomes a circle 22 on the projected encounter plane 18. Likewise, the covariance ellipsoid becomes an ellipse 24 as depicted in FIG. 2.
The relative velocity vector 16 (decoupled dimension) is associated with the time of closest approach (TCA). The conjunction assessment here is concerned with cumulative probability over the time it takes to span the n-σ shell, not an instantaneous probability at a specific time within the shell. Along this decoupled dimension, integration of the probability density across the shell produces a number very near unity, meaning the close approach will occur at some time within the shell with near absolute certainty. Thus the cumulative collision probability is reduced to a two-dimensional problem in the encounter plane 18 that is then multiplied by the decoupled dimension's probability. By rounding the latter probability to one, it is eliminated from further calculations. This projection results in a double integral.
The resulting two-dimensional probability equation in the encounter plane 18 is given as
                    P        =                              1                                          2                ·                π                ·                σ                            ⁢                                                          ⁢                              x                ·                σ                            ⁢                                                          ⁢              y                                ·                                    ∫                              -                OBJ                            OBJ                        ⁢                                          ∫                                  -                                                                                    OBJ                        2                                            -                                                                        (                          x                          )                                                2                                                                                                                                                        OBJ                      2                                        -                                                                  (                        x                        )                                            2                                                                                  ⁢                                                exp                  ⁡                                      [                                                                  (                                                                              -                            1                                                    2                                                )                                            ·                                              [                                                                                                            (                                                                                                x                                  +                                  xm                                                                                                  σ                                  ⁢                                                                                                                                          ⁢                                  x                                                                                            )                                                        2                                                    +                                                                                    (                                                                                                y                                  +                                  ym                                                                                                  σ                                  ⁢                                                                                                                                          ⁢                                  y                                                                                            )                                                        2                                                                          ]                                                              ]                                                  ⁢                                  ⅆ                  y                                ⁢                                                                  ⁢                                  ⅆ                  x                                                                                        (        2        )            where OBJ is the combined object radius, x lies along the minor axis, y lies along the major axis, xm and ym are the respective components of the projected miss distance, and σx and σy are the corresponding standard deviations. The four methods discussed below approximate Equation 2 numerically (Foster, Patera, Alfano) or analytically (Chan).
Foster (see Foster, J. L., and Estes, H. S., “A Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Vehicles,” NASA/JSC-25898, August 1992) derived a collision probability model using polar coordinates in the encounter (U-W) plane where R0 and φ define the combined object center's location, OBJ is the combined object radius, σu and σw are the principal axes standard deviations, and r and θ define the relative spatial position of the segmented object.
                    P        =                              1                                          2                ·                π                ·                σ                            ⁢                                                          ⁢                              u                ·                σ                            ⁢                                                          ⁢              w                                ·                                    ∫              0              OBJ                        ⁢                                          [                                                      ∫                    0                                          2                      ·                      π                                                        ⁢                                                                                    exp                        ⁡                                                  [                                                                                                                    -                                1                                                            2                                                        ·                                                          [                                                                                                                                    (                                                                                                                                                                                                        R                                            0                                                                                    ·                                                                                      sin                                            ⁡                                                                                          (                                              ϕ                                              )                                                                                                                                                                      -                                                                                  r                                          ·                                                                                      sin                                            ⁡                                                                                          (                                              θ                                              )                                                                                                                                                                                                                                                  σ                                        ⁢                                                                                                                                                                  ⁢                                        u                                                                                                              )                                                                    2                                                                +                                                                                                      (                                                                                                                                                                                                        R                                            0                                                                                    ·                                                                                      cos                                            ⁡                                                                                          (                                              ϕ                                              )                                                                                                                                                                      -                                                                                  r                                          ·                                                                                      cos                                            ⁡                                                                                          (                                              θ                                              )                                                                                                                                                                                                                                                  σ                                        ⁢                                                                                                                                                                  ⁢                                        w                                                                                                              )                                                                    2                                                                                            ]                                                                                ]                                                                    ·                      r                                        ⁢                                                                                  ⁢                                          ⅆ                      θ                                                                      ]                            ⁢                              ⅆ                r                                                                        (        3        )            
In Foster's numerical implementation, the angle θ step size is 0.5° and the radius r step size is OBJ/12. This model is currently used by NASA to assess on-orbit risk for ISS and Shuttle missions. It can also be found in The Aerospace Corporation's Collision Vision Tool. Solution accuracy is degraded when the object radius is smaller than the miss distance but larger than the standard deviation of the minor axis. Within the accuracy bounds of currently available orbital data, it is reasonable to assume that these theoretical cases are highly unlikely.
Patera (see Patera, R. P. “General Method for Calculating Satellite Collision Probability,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4, July-August 2001, pp. 716-722) developed a mathematically equivalent model to Equation 2 as a one-dimensional line integral where r is the distance to the hardbody perimeter and θ is the covariance-centric angular position. The probability density is symmetrized enabling the two-dimensional integral to be reduced to a one-dimensional path integral, resulting in the expression
                    P        =                                            -              1                                      2              ·              π                                ·                                    ∮              ellipse                        ⁢                                          exp                ⁡                                  (                                                            -                      α                                        ·                                          r                      2                                                        )                                            ⁢                              ⅆ                θ                                                                        (                  4          ⁢          a                )            if the miss distance exceeds the combined object radius and
                    P        =                  1          -                                    1                              2                ·                π                                      ·                                          ∮                ellipse                            ⁢                                                exp                  ⁡                                      (                                                                  -                        α                                            ⁣                                              ·                                                  r                          2                                                                                      )                                                  ⁢                                  ⅆ                  θ                                                                                        (                  4          ⁢                                          ⁢          b                )            if the combined object radius exceeds the miss distance. Computation of the α term and Equation 4's numerical implementation involves coordinate rotation, scaling, and trigonometric functions. In a subsequent Engineering Note (see Patera, R. P. “Calculating Collision Probability for Arbitrary Space-Vehicle Shapes via Numerical Quadrature,” Journal of Guidance, Control, and Dynamics, Vol. 28, No. 6, November-December 2005, pp. 1326-1328), Patera switched the integration variable to be object-centric and employed a series expansion when r was very small. These changes overcame occasional computational difficulties of the original method and also resulted in substantially fewer iterations to achieve a given level of accuracy. This method is currently employed in The Aerospace Corporation's Collision Vision Tool.
The present inventor Alfano (see Alfano, S. “A Numerical Implementation of Spherical Object Collision Probability,” Journal of the Astronautical Sciences, Vol. 53, No. 1, January-March 2005, pp. 103-109) developed a series expression to represent Equation 2 as a combination of error (erf) functions and exponential terms. In the encounter plane 18, the combined object center's location is (xm, ym) with associated standard deviations σx and σy and combined object radius OBJ. The series expression is given as
                    P        =                                            OBJ              ·              2                                                                                            8                    ·                    π                                                  ·                σ                            ⁢                                                          ⁢                              x                ·                n                                              ·                                    ∑                              i                =                0                            n                        ⁢                                          [                                                                                                                              erf                          [                                                                                    [                                                              ym                                +                                                                                                                                            2                                      ·                                      OBJ                                                                        n                                                                    ·                                                                                                                                                    (                                                                                  n                                          -                                          ⅈ                                                                                )                                                                            ·                                      ⅈ                                                                                                                                                                  ]                                                                                      (                                                              σ                                ⁢                                                                                                                                  ⁢                                                                  y                                  ·                                                                      2                                                                                                                              )                                                                                ]                                                +                                                                                                                                                erf                        [                                                                              [                                                                                          -                                ym                                                            +                                                                                                                                    2                                    ·                                    OBJ                                                                    n                                                                ·                                                                                                                                            (                                                                              n                                        -                                        ⅈ                                                                            )                                                                        ·                                    ⅈ                                                                                                                                                        ]                                                                                (                                                          σ                              ⁢                                                                                                                          ⁢                                                              y                                ·                                                                  2                                                                                                                      )                                                                          ]                                                                                            ]                            ·                              exp                [                                                      -                                          [                                                                                                    OBJ                            ·                                                          (                                                                                                2                                  ·                                  ⅈ                                                                -                                n                                                            )                                                                                n                                                +                        xm                                            ]                                                                                                  2                      ·                      σ                                        ⁢                                                                                  ⁢                                          x                      2                                                                      ]                                                                        (                  5          ⁢                                          ⁢          a                )            
The method then breaks the series into m-even and m-odd components and makes use of Simpson's one-third rule. An expression to determine a sufficiently small number of terms is given as
                    m        -                  int          (                                    5              ·              OBJ                                      min              (                                                σ                  ⁢                                                                          ⁢                  x                                ,                                  σ                  ⁢                                                                          ⁢                  y                                ,                                                                            xm                      2                                        +                                          ym                      2                                                                                  )                                )                                    (                  5          ⁢                                          ⁢          b                )            with a lower bound of 10 and upper bound of 50. This method is currently implemented in Satellite Tool Kit (STK®) from Analytical Graphics, Inc. of Exton, Pa.
Chan (see Chan, K., “Improved Analytical Expressions for Computing Spacecraft Collision Probabilities,” AAS Paper No. 03-184, AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, 9-13 Feb. 2003) developed a series expression as an analytical approximation to Equation 2. It is based on transforming the two-dimensional Gaussian probability density function (PDF) to a one-dimensional Rician PDF and using the concept of equivalent areas. In the encounter plane, the combined object radius is OBJ, centered at (xm, ym) with associated standard deviations of (σx, σy). The series expression is
                              P          =                                    exp              ⁡                              (                                                      -                    v                                    2                                )                                      ·                                          ∑                                  m                  =                  0                                ∞                            ⁢                              [                                                                            v                      m                                                                                      2                        m                                            ·                                              m                        !                                                                              ·                                      (                                          1                      -                                                                        exp                          ⁡                                                      (                                                                                          -                                u                                                            2                                                        )                                                                          ·                                                                              ∑                                                          k                              =                              0                                                        m                                                    ⁢                                                                                    u                              k                                                                                                                      2                                k                                                            ·                                                              k                                !                                                                                                                                                                          )                                                  ]                                                    ⁢                                  ⁢        where                            (                  6          ⁢                                          ⁢          a                )                                u        =                                                            OBJ                2                                            σ                ⁢                                                                  ⁢                                  x                  ·                  σ                                ⁢                                                                  ⁢                y                                      ⁢                                                  ⁢            v                    =                                                    xm                2                                            σ                ⁢                                                                  ⁢                                  x                  2                                                      +                                          y                ⁢                                                                  ⁢                                  m                  2                                                            σ                ⁢                                                                  ⁢                                  y                  2                                                                                        (                                            6              ⁢                                                          ⁢              b                        &                    ⁢                                          ⁢          6          ⁢          c                )            
This expression has the added benefit of being easily differentiated for other types of probability analysis. This model is currently implemented in the Satellite Tool Kit from Analytical Graphics, Inc. Solution accuracy is degraded when the object radius is larger than one-tenth the miss distance. This is expected because Chan used an equivalent-area approximation.
However, the assumption of linear relative motion may not be valid in all cases. Chan (see Chan, K., “Spacecraft Collision Probability for Long-Term Encounters,” AAS Paper No. 03-549, AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Mont., 3-7 August, 2003), Patera (see Patera, R. P. “Satellite Collision Probability for Nonlinear Relative Motion,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 5, 2003, pp. 728-733), Alfano (see Alfano, S., “Addressing Nonlinear Relative Motion For Spacecraft Collision Probability,” AIAA Paper No. 2006-6760, 15th AAS/AIAA Astrodynamics Specialist Conference, Keystone, Colo., Aug. 21-24, 2006), and McKinley (see McKinley, D. P., “Development of a Nonlinear Probability Collision Tool for the Earth Observing System,” AIAA Paper No. 2006-6295, 15th AAS/AIAA Astrodynamics Specialist Conference, Keystone, Colo., Aug. 21-24, 2006) each proposed different methods for calculating collision probability for such instances. Nonlinear motion is typically associated with long-term encounters, which imply the covariance can no longer be assumed static. The collision tube will not be straight, invalidating the simple dimensional reduction used for linear motion. The size of the n-σ shell must also be carefully considered, especially if the relative motion reverses direction during the encounter. The cumulative collision probability P is found by integrating the three-dimensional, Gaussian, relative position density over the volume V (collision tube) that is swept out by the combined hardbody of the two space objects over a specified time interval (ti, tf)
                    P        =                              1                                                            8                  ·                                      π                    3                                                              ·                              σ                x                            ·                              σ                y                            ·                              σ                z                                              ⁢                                    ∫                              ∫                ∫                                                    V              (                                                t                  i                                ,                                  t                  f                                            )                                ⁢                      exp            ⁡                          [                                                                                                                                            -                                                      x                            2                                                                                                    2                          ·                                                                                    (                                                              σ                                x                                                            )                                                        2                                                                                              +                                                                        -                                                      y                            2                                                                                                    2                          ·                                                                                    (                                                              σ                                y                                                            )                                                        2                                                                                              +                                                                                                                                                          -                                                  z                          2                                                                                            2                        ·                                                                              (                                                          σ                              z                                                        )                                                    2                                                                                                                                ]                                ⁢                      ⅆ            x                    ⁢                      ⅆ            y                    ⁢                      ⅆ            z                                              (        7        )            The probability density in the bracketed section is conveniently represented in the diagonal frame of the position-error covariance matrix. The definition of the integration volume V(ti, tf) is the most complicated aspect of evaluating this expression.
The previously described linear methods for computing satellite collision probability can be extended to accommodate nonlinear relative motion in the presence of changing position and velocity uncertainties. For linear relative motion, the probability along the relative velocity vector (collision tube) is unity and is conveniently removed from the calculations. For nonlinear motion, that dimension must be reintroduced. This can be simply done by breaking the collision tube into small sections, computing the probability associated with each section, and then summing. To accomplish this, an (orbit) propagator is needed that can propagate object (satellite) position, velocity, and associated covariances. The propagator must be of sufficient accuracy to meet user requirements.
The general method of adjoining tubes begins with object position and velocity data at the time of closest approach. Propagation is done forward/backward in time until a user limit is reached. The limit can be based on a standard deviation threshold (3σ was mentioned by Patera and an upper limit of 8.5σ recommended by Chan) or a specified time (such as one half an orbital period). For each time step the tube sections should be sufficiently small enough so that, over the interval, the relative motion can be assumed linear and the covariance constant. For each section, a two-dimensional probability is computed as previously described for linear motion by projecting the combined object shape onto a plane perpendicular to the relative velocity. In addition, a one-dimensional probability is computed along the relative velocity vector by determining the component position from the mean at each end of the tube and then dividing by the standard deviation for that axis, thus producing each endpoint's Mahalanobis distance (see Alfano, S., “Addressing Nonlinear Relative Motion For Spacecraft Collision Probability,” AIAA Paper No. 2006-6760, 15th AAS/AIAA Astrodynamics Specialist Conference, Keystone, Colo., Aug. 21-24, 2006). The product of these probabilities yields the sectional probability. All sectional probabilities are summed until the time and/or sigma limit is reached. This approach differs from Patera's original work in that the symmetrized space is time-invariant resulting in a new derivation of the path integral. The probability of each cylinder is determined by multiplying the two-dimensional linear probability by the sectional (relative velocity axis) probability; the user may choose any of the linear probability models previously described in the literature.
The tubes have no gaps when dealing with linear relative motion. For such cases, the nonlinear results will match the linear probability for constant covariance and spherical objects. As seen in FIG. 3, nonlinear motion causes gaps 32 and overlaps 34 where the tube sections 36 meet. If the relative motion track 38 bends towards the covariance ellipsoid center, then the overlapping sections 34 will occur in regions of greater probability density with the gaps 32 occurring in regions of lesser probability density. Although the gap and overlapping volumes are almost equal, the resulting summation causes an over inflation of the probability. If the relative motion track 38 bends away from the covariance ellipsoid center, then the probability for cylindrical tubes will be underestimated because the gap 32 is in a region of higher probability density. The amount of error will vary based on the degree of bending/overlap relative to probability density as well as the rate of covariance growth during the encounter time.
There are several choices the user should carefully consider when implementing this method. The limits and time step must be selected to ensure adequacy for the intended analysis. For a very large time limit and cyclical relative motion it is possible to retrace the same path through the covariance space. An example would be one satellite circling the other in formation for hundreds of revolutions. The collision tube would continually trace over itself; if care is not taken, the single revolution probability could be summed hundreds of times. To avoid this, it is suggested that the total time limit not exceed one half of an orbital period or that subsequent retracing be recognized and suitable adjustments made to the calculation. A large time step can also cause errors if the sectional motion is not sufficiently linear or the sectional covariance is not sufficiently constant. A simple test for sufficiency is to halve the time step and repeat the analysis. If the probability differences are within the user's tolerance, then the time step is adequate.
Three-dimensional position and velocity data of each object, as well as their 6×6 covariance matrices, are required. Although not necessary, this work assumes all starting data to be at the point of closest approach in the Earth Centered Inertial (ECI) frame. Suitable incremental limits should be set for the time step, maximum acceptable angle (angular limit) between adjoining tubes, and maximum change in long-axis sigma for any tube. Additionally, the user must specify the computational stopping condition in terms of time limit and/or encounter region.
To compute the sectional probability of each tube, all data is propagated for the given time step. If the angular difference between the previous and current relative velocity vectors exceeds the angular limit, the time step is halved and this process repeated. A coordinate transformation is accomplished to align the z axis with the relative velocity vector. The one-dimensional, z-axis, Mahalanobis distances of the cylinder endpoints (Mi, Mf) are used to compute long-axis probability P1d from
                              P                      1            ⁢                                                  ⁢            d                          =                                                      1              2                        ·                          (                                                erf                  (                                                            M                      f                                                              2                                                        )                                -                                  erf                  (                                                            M                      i                                                              2                                                        )                                            )                                                                    (        8        )            
If the endpoint differences should exceed the maximum change in long-axis sigma then the time step is halved and this process repeated. Projection of the collision tube onto the encounter plane, defined by the x-y axes, produces the necessary information to generate two-dimensional collision probability using whatever method the user chooses. The one- and two-dimensional probabilities are then multiplied to determine the sectional collision probability. This process is repeated until a final limit is reached.
A 3σ encounter shell may be insufficient for some cases where the relative trajectory reverses itself. Consider the linear and nonlinear motion shown in FIG. 4A. The nonlinear relative motion is such that the trajectory 42 exits and reenters the 3σ encounter shell 44 both before and after the close approach point. The positional history represented by the Mahalanobis distance 46 is plotted versus time in seconds in FIG. 4B. With a final calculation limit of 3σ, the nonlinear probability is 0.000747. Expanding the limit to 10σ, the nonlinear probability is 0.00111. In this case, the 3σ limit did not fully capture the encounter. As a point of reference, the linear probability is 0.000469.
Patera presented a nonlinear toroidal case for testing. A circular, relative trajectory 52 is chosen with a spherical hardbody radius and symmetric covariance ellipsoid. The object creates a torus 54 as it follows the circular trajectory 52. The exact solution to collision probability was derived by members of The Aerospace Corporation as:
                              p          =                                    2              σ                        ·                                          2                π                                                    ⁣                  ·                      exp            ⁡                          [                                                -                                      (                                                                  R                        2                                            +                                              r                        2                                                              )                                                                    2                  ·                                      σ                    2                                                              ]                                ·                                    ∫              0              r                        ⁢                                          sinh                (                                                      R                    ·                                                                                            r                          2                                                -                                                  x                          2                                                                                                                          σ                    2                                                  )                            ⁢                                                          ⁢                              ⅆ                x                                                                        (        9        )            where σ is the standard deviation, R is the radius of the torus, and r is the cross-sectional radius as shown in FIG. 5.
The collision tube is more closely represented as the number of cylinders increases. With σ set to one, R set to one, and r set to 0.3, Eq. 9 produces a probability of 0.066144. The number of adjoining cylinders was varied from 4 to 300 to assess convergence behavior as displayed in FIG. 6 with the label “Adjn Tubes”. Representing the torrus with 300 adjoining cylinders, the probability value was 0.06765. This is an overestimate of 2.3% and is in agreement with Patera. Because the tube bends towards the origin, the cylinders will overlap in regions of greater probability density and cause an overestimation. The collision probability calculation does not require a large number of tubes. Even with an angle limit of 5 degrees, the probability is 0.06767, suggesting that the test for nonlinearity given by Chan may be relaxed in certain cases.
The gaps and overlaps created by adjoining right circular cylinders can be reduced by sectioning the cylinder into component pieces. The gaps and overlaps of all the sections will be considerably smaller than the unsectioned cylinder. For Foster's method, the cylinder is sectioned into 12 radial segments and 720 angular segments. Patera (see Patera, R. P., “Collision Probability for Larger Bodies Having Nonlinear Relative Motion,” Journal of Guidance Control and Dynamics, Vol. 29, No. 6, November-December 2006, pp. 1468-1471) found that by using his methods, five radial segments and 20 angular segments are adequate for the cases he examined.
What would be useful is a non-polar form of sectioning that has the added benefit of easily representing any object shape.